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Optical Design

  • Page ID
    75541
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    Single grating, Single detector

    We will use this Czerny-Turner spectrometer with an exit slit and photomultiplier. The most common single-channel system is a Czerny-Turner spectrometer with matched entrance and exit slits and a photomultiplier detector. The grating and two mirrors are arranged so that the entrance slit is imaged onto the exit slit, light from the entrnace slit is collimated onto the diffraction grating, and the grating can be rotated to an angle θ, such that when θ = 0, the grating functions in zero order, passing all wavelengths. The layout and grating equation are shown in the following figure; in this geometry, the grating equation can be shown to be nλ = 2d sin θ cos β. For slit width w, the range of wavelengths observed, Δλ = wd cos β/(nf) = wd cos (θ+β)/(nf), with f the focal length of the spectrometer camera mirror.

    CT_dwg.gif

    Exercise

    Suppose you have a 1200 groove per mm grating (a common groove density). You can put the grating into a spectrometer with focal length 100 mm, one with focal length 250 mm, one with focal length 1000 mm, or one with focal length 2000 mm. If you want to look at absorption of a species in solution at a wavelength of about 450 nm, where resolution of 1 nm is adequate, what focal length, order, and slit width do you choose? If you want to look at atomic emission at a similar wavelength where resolution of 0.1 nm is mandatory and resolution of 0.01 nm is desirable, what order, slit width, and focal length do you choose? Answer

    Concave grating, Multiple detector

    The Paschen-Runge Direct Reading Spectrometer. Consider a spherical mirror with radius of curvature R. A light source at the center of the sphere would emit light that would strike each point on the surface of the spherical mirror normal to the surface and retro-reflect back to the source. Since the "paraxial approximation" for focusing (the focusing behavior for objects close to the axis of an optical system) relates that 1 divided by the distance from source to optic plus 1 divided by the distance from optic to focus equals 1 divided by the focal length of the optic, or 1/S1+1/S2=1/f, then 2/R = 1/f for a spherical mirror (S1 = S2 = R). What happens if in place of the spherical mirror we put a spherical grating, placing an entrance slit at a distance R from the grating, aligned with the grating's normal or axis? The grating equation still works, but with entrance angle α = 0. nλ = d sin β (we cheated on the sign convention -- it's convenient to choose n and β to have the same sign.). The focal geometry is shown at the following link.

    RolandPaschen-small.JPG

    The various wavelengths focus on a Roland circle (the diameter of which is anchored at the entrance aperture and grating center). In the days of photographic film detectors, one could simply wrap flexible film along this curve and obtain a spectrum with ultraviolet wavelengths recorded near the end of the film close to the entrance aperture, longer wavelengths farther along the focal curve, and distance along the film from the entrance slit = 2βR, where β is in radians. If one wished to use photoelectric detection, slits could be positioned on the Roland circle at the appropriate position for the desired wavelength. If an array of photomultipliers was used, the result was a "direct reader," a spectrometer that could simultaneously observe N wavelengths with N slits and N photomultipliers. There were several problems:

    1. The focus was highly astigmatic. While a line at short wavelength and small angle β would have the same height as the entrance slit, lines at long wavelengths would be much longer. This reduced sensitivity by spreading a given number of photons over a greater image height or area than was the case for short wavelengths.
    2. No two lines closer together than the width of slits and their holders could be observed.
    3. Any temperature changes adjusted the position of the slits in such a manner that the central observed wavelength drifted. If an instrument was calibrated at one temperature and used at a different temperature, the observed wavelength was not the desired one.
    4. Because the focal "plane" is curved, modern charge-coupled arrays or diode arrays were difficult to use.

    Fortunately, holography can be used to make gratings whose focal plane is in fact close to planar and with drastically-reduced astigmatism. Many inexpensive low-resolution diode array spectrometers use an aberration-corrected Paschen-Runge mount for measurement. In the world of elemental analysis, the Paschen-Runge arrangement dominated the multiple wavelength simultaneous measurement world from the early 1960s until CCD/echelle systems became common around the turn of the 21st century. It is still commonly employed, especially if the time-resolving capabilities of photomultipliers are useful. When is that? When the signal-to-background ratio varies rapidly with time. For laser-induced breakdown or spark emission spectrometry, where the early stages of emission are rich in continuum but later stages have greatly diminished continuum, photomultiplier detection is still common and thus so is the multiple slit, multiple detector Paschen-Runge system. See e.g. Thermo Electron's spark emission instrument. In Europe, Spectro Inc.makes a Paschen-Runge slit and PMT system.

    Plane grating, Multiple detector

    If a Czerny-Turner or Ebert-Fastie spectrometer has an exit slit in its focal plane, only a single, narrow band of wavelengths will be observed passing through the slit. If instead of a slit, one puts a CCD in the focal plane, a range of wavelengths can be observed. Suppose one is using a 1 m focal length Czerny-Turner spectrometer with a 1200 groove per millimeter grating. Using the dispersion relationship for such instruments, it can be shown (try it!) that the dispersion is approximately 0.83 nm/mm. Suppose the CCD has 25 µm pixels and is 1000 × 1000 pixels. Then each pixel will observe (nominally) 0.025 mm times 0.83 nm/mm or 0.021 nm, and the CCD as a whole will span 25 mm (physical distance) or 21 nm (spectral window). One must be a bit careful in considering the nuances of the situation. The focal plane of a Czerny-Turner system is not perfectly flat, while the CCD chip is very close to being a Euclidian plane. Field curvature, astigmatism, and coma will all perturb the line shape, making resolution poorer than a simple dispersion calculation might indicate. The image of the entrance slit will be curved in the focal plane, so that lines will appear to be circular arcs if the entrance slit is straight. If the entrance slit is illuminated only at a single point, astigmatism will render the spectrum in the tangential focus (the plane in which the spectral image is formed) as a line of non-negligible height. Just as for the Paschen-Runge mount, astigmatism spreads the light out in space, rendering signal amplitude on a single pixel lower than would be the case for perfect imaging. Examples can be found at Intevac and Bruker/Chromex.

    Plane grating, Crossed with prism or grating, Multiple detector

    The cross-dispersion echelle spectrometer is the state of the art in elemental analysis. Examples include Andor/Mechelle, Teledyne/Leeman, and Varian. The echelle grating, operating at a blaze angle of 75° or thereabouts, disperses the UV and visible regions of the spectrum in ~ 50th to 150th order. Cross dispersion by a non-blazed grating or prism separates the orders. The dispersion values are chosen so that all lines of interest fall on a CCD or other array detector with sufficient dispersion that physical line shape, not instrument dispersion, limits resolution. The patent on the Varian instrument is freely available: U.S. patent 5,596,407. If you have Quicktime on your computer, you can see the graphics. Alternatively, if you have a subscription to Applied Spectroscopy, go to 45(3), 334-346 (1991) for a full description of how one such system is engineered.

    A somewhat simplified situation can be described by modeling the echelle as operating in Littrow mode, so α = β = 70° at the center of every order and nλ = 2d sin 70°. Suppose one wants to have 325 nm right in the center of an order. Then n × 325 nm = 2 d × 0.93969. d = n × 172.93 nm. One equation, two variables, what can one do? Choose the desired dispersion. For 25 µm pixels, one may wish 0.003 nm resolution (10 pm is approximately the width of atomic emission lines, and this way we have 3 pixels across each line). That works out to 0.003 nm/0.025 mm or 0.12 nm/mm = d cos β/nf. d/nf = (0.12 nm/mm)/cos 70° or d/nf = 0.3509 nm/mm. d/n = 172.93 nm, so 172.93/f = 0.3509, f = 606.74 mm. This is not a standard focal length for a mirror. 600 mm is a sensible value. But we still have no way to separate n from d. One might check to see what d spacings are in grating catalogs. The Richardson Grating Laboratory, now part of Newport Corp., has a catalog of possible gratings. Groove densities for gratings with 70° blaze include 27, 44.41, 158, and 316 grooves/mm. Let's tabulate the behavior with which these correspond, both at 325 nm and at 200 nm (we'll need to know how many orders span this range momentarily).

    Grooves/mm

    d (nm)

    n (325 nm, rounded)

    n (200 nm, rounded)

    27

    37037

    214

    348

    44.41

    22518

    130

    212

    158

    5329

    31

    50

    316

    3164.6

    18

    30

    We thus can have anywhere from (348-214) = 134 to (30-18) = 12 orders to cover the portion of the UV spectrum in which a majority of atomic emission lines are found. The cross-dispersion has to be sufficient so that adjacent orders don't overlap, confounding identification of spectral lines. If we want to cover half the CCD with 200 to 325 nm, we need to know how far apart adjacent orders will be and thus how many pixels must separate each order. While the free spectral range, the range of wavelengths in a given order, varies from order to order (higher orders have shorter free spectral range), we can get an initial guestimate by ignoring the change. Taking order difference divided by wavelength range, we see that average free spectral range varies from 125 nm/134 orders to 125 nm/12 orders. That's anywhere from 1 to 10 nm per order. But wait! We know the dispersion near 325 nm! It's 0.003 nm/pixel, and the typical CCD has 1000 pixels across! That means we can't have more than 3 nm across a single order at 325 nm or we'll lose information off the edge of the CCD. That means we need an average of less than 3 nm per order. If we pluck 2.5 nm/order out of thin air, we're looking for 125 nm/2.5 nm/order = 50 orders. NONE of our choices match this! Fortunately, now that we have a better idea of what free spectral range and number of orders we desire, we can go back to the grating catalog and choose a blaze NEAR (but not equal to) 70°, and something between 45 and 158 grooves/mm to get something that will work. This iterative approach (choose a parameter, work out its implications, refine the constraints, and repeat until the system is workable) is typical of how one chooses instrumental parameters.


    This page titled Optical Design is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Alexander Scheeline & Thomas M. Spudich via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.